Probability is hard

Our 20 week ultrasound revealed a healthy, growing baby, but there was one worrisome characteristic. Baby had a skin fold on the back of its neck whose thickness was enough to indicate a correlation with an increased chance of Down’s syndrome.

Jeff and I considered not getting any screening — even with this indicator, the chances of Down’s syndrome are pretty low — but in the end, we decided to do a maternal blood screen. Unlike an amniocentesis, this test does not provide a certain answer, but it also doesn’t post the very small but non-zero risk to baby that the amnio does. A blood screen is also more comfortable for me. 
But it does have its drawbacks. The screening has a 99% detection rate and a 0.1% false positive rate[1]. That sounds pretty good, but running the numbers, you’ll see why probability is non-intuitive. 
Given my age and the skin fold measurement, the baseline probability for Down’s is between 1:100 and 1:150. So out of 10,000 babies in this group[2]:
  • 100 will have Down’s syndrome. Of this 100, 99 will be detected by the screen. 1 will be missed.
  • 9900 will not have Down’s syndrome. Of this, 9.9 will be incorrectly be above the risk threshold for Down’s syndrome in the screen. The other 9890.1 will be correctly diagnosed as not having Down’s.
Combining these, if the screen comes back as a “no” there is only a 1:9891.1 chance that baby has Down’s. But a “yes” on the screen indicates a ~90% chance of Down’s and an ~10% chance of not. In other words, a “no” leaves you pretty confident, but a “yes” still has a lot of ambiguity. 
All-in-all, the screen seemed like a better option for us, but these decisions aren’t easy. 
By the way, the results came back negative. Yay!
[1] As claimed by the test provider, but these sorts of measurements are themselves rather difficult to do well.
[2] By the way, this may be completely wrong. Probability was not my strong suit in mathematics. Math with numbers is hard.